This first image shows the monotone polygon I designed with horizontal lines corresponding to line sweep.
In this image I have triangulized my polygon according to the three cases with corresponding colors:
Created with Cinderella
1. The current vertex and the previous vertex are on separate chains.
2. The current vertex and the previous are on the same chain making a convex angle with the next vertex.
3. The two vertices are on the same chain but with a concave angle with the next vertex. Point is put on stack.